Extreme Narrow Escape: Shortest paths for the first particles among n to reach a target window

What are the paths associated with the fastest Brownian particles that reach a narrow window located on the boundary of a microdomain? Although the distribution of the fastest arrival times has been well studied in dimension 1, much less is known in higher dimensions. Based on the Wiener path-integral, we suggest that the paths of the fastest particle are concentrated near the shortest paths that minimize the energy-action. Stochastic simulations confirm the present result when an obstacle is positioned between the source point and a narrow window. To conclude paths associated with the fastest arrival times differ significantly from the ones of mean properties of Brownian motions, associated to mean first passage times of a single particle. These extreme properties should be considered instead of the classical Smoluchowski's rate of chemical reactions, because the statistics of the extreme for many copies of the same molecule changes the time scales of activation in cellular domain.